Exploring
chords in the Wilson CPS sets

The Wilson CPS sets provide
a wonderful opportunity for exploring the various just intonation
generalisations of minor / major chords. CPS stands for "combination
product set" and Erv Wilson who devised them is a renowned
theorist and inventor of scales.

The models on this page are
in VRML. There are good viewers for these on the p.c., but
unfortunately, I haven't heard of any that show these models okay
on the mac. (especially, beware of the Cosmo Player beta for the
mac - while the Cosmo player is one of the best viewers for the P.c.,
I've been told that the mac beta can erase your address book when
you install it!! If you want to try it, back up your address book
first.).

Two exellent VRML viewers
for the p.c: Cosmo Player, and
Cortona. It is
harder to find a good one for the Mac - the Cosmo player beta
isn't recommended as I've heard of people having problems with it.
Cortona have recently made a release of their browser for the
Mac, so maybe this is the one we've been waiting for??

The idea of a 2 3 5 7 hexany
is that you take all pairs of numbers from 2 3 5 7 and multiply
them together:

2*3
2*5 2*7 3*5 3*7 5*7

That gives six numbers. You
can then place them on the vertices of an octahedron, and if you
wish, octave reduce them. The octahedron used this way is known
as a hexany because it has six vertices.

(program uses 2 instead of 1,
but up to octave reduction, it's the same).

To make this into a
conventional scale, note that 1*1 isn't in it, so we don't yet
have a 1/1.

So choose one of these notes
as the 1/1, say, 5*7. Divide all the notes by 5*7 and you get:

1/1
8/7 6/5 48/35 8/5 12/7 2/1 (up
to octave reduction).

The interesting thing about
this scale is that if you put the notes on the vertices of an
octahedron, and if each face plays the three notes for its
vertices as a chord, then all the chords are consonant triads. So
it is a great way of exploring triads and making a scale with
many interesting triads in it.

It is best to wait for
everything to load before you start clicking on the vertices -
this one shouldn't take that long to load.

Click on any sphere above
the centre of a face to hear the triad. Click on any vertex to
hear the note, and on any of the spheres in the middle of the
edges to hear a diad.

If your VRML browser has a
choice of Study mode, you will find this the most useful for
looking at these models, as it keeps the model centred. You may
also be able to use the drop list of viewpoints which I've
provided for the model.

You need to wait for sound
to stop before clicking on another note / chord in the model. If
you don't do that, you may find a note or chord stops sounding,
and have to reload the model again in order to hear it.

Any just intonation triad
can be written in two ways, e.g. 1 3 5 can
be written as 15/15 15/5 15/3, = 1/3 1/5 1/15.

If a triad is simplest with
the numbers on the top (ignoring any powers of two), it is an otonal triad - expressed most simply using the overtone series.
If simplest with them on the bottom, its a utonal triad - expressed most simply using the undertone series.

The major chord 1 5/4 3/2 is an example of an otonal triad, and theminor chord 1 2/3 4/5 is an example of a utonal triad. The hexany has
both of these, and also has other otonal and utonal chords such
as 1/1 3/2 7/4 and 1/1 2/3 4/7.

Otonal and utonal chords
with the same factors are on opposite faces of the hexany.

The hexany is a common
component of larger CPS sets.

Now lets look at the dekany,
which is what you get if you add one more factor:

Add another factor, and you
need a fourth space dimension at right angles to the three we are
familiar with.

It is not the dimension of
time though, just another space dimension, if one can imagine
such a thing, or more likely, fail to imagine it!

There are records of people
who have said they can get an inkling of the idea of a fourth
space dimension, and even solve problems in 4D by imagining the
shapes in periods of concentration. But most are happy to view
projections of it, and just rely on the maths to get it right.

What one can do is to make a
projection into a smaller number of dimensions.

We do this whenever we draw
a 3D figure on a sheet of paper. In the same way, one can draw a
4D figure in 2D, or indeed, in 3D.

It
will take a while to load the sound clips, as they all need to be transferred, and it is
best to wait for them all to load before you start clicking on
them.

Alternatively, you can use
the zip [200KB] 278
files. Unpacks to 971 Kb.

The zip has this one, the
next one, and the dekanies using the factor 9 instead of 11. (Because
of number of files in it, though the total size of all the files
is 971 KB, it will actually use 9 - 10 Mb if your hard disk has
32K clusters ).

Click on anything and it
will sound a note or a chord. The utonal triads are shown as
transparent triangles, and the otonal ones are solid. You can
click on the otonal triangles to hear those.

This model uses a
particularly symmetrical projection that I adapted from one that
Paul Erlich made. It's a perspective 3D view on a 4D solid.

The outer hexany of the
model is apparently larger and outside - that's because it is
nearer in the fourth spatial dimension, and one is looking
through it to the tetrahedron gap in the centre. The tetrahedron
in the centre is smaller because it is further away. All the
triangles you see in this entire model are actually part of the
outside of the four dimensional shape!

The octahedron and
tetrahedron are nested neatly within each other because one is
looking at it from directly opposite the outer octahedron in the
fourth space dimension. Compare the method of drawing a cube as
two concentric squares joined to each other by radial diagonal
lines. This symmetrical view makes it easier to look at in 3D
perspective.

Choosing two factors at a
time out of a list of six, such as 1 3 5 7 11 13, gives the
pentadekany, which is made out of six overlapping 2)5 dekanies.

You can also select any four
factors at a time to get another pentadekany made out of six
overlapping 3)5 dekanies. The pentadekany is a five dimensinoal
figure, so hard clearly to show in a 3D projection.

This web page links to some
models of the constituent dekanies of the 2)6 pentadekany, and a
3D model that shows most (but not all) of the chords of the
complete figure. The two versions of the page show the same
chords, but one is played as unison chords on a violin, and the
other as broken chords on the 'cello midi voice.

The 2)6 pentadekany has
otonal pentads which you get by selecting one factor and the 4)6
pentadekany has utonal pentads which you get by skipping one
factor. I don't yet have a model for the 4)6 pentadekany.

The Eikosany is even richer
in chords than the two dekanies, and by selecting any one of the
factors you will get the 2)5 dekany, and by skipping one of them
you get a 3)5 dekany.

So, the Eikosany is made out
of twelve overlapping dekanies. Imagine how many chords that
makes!

I don't have a model of one,
and it may be a bit too complex for a complete 3D model. However
the constituent dekanies are all the ones on the Pentadekany page.
The Eikosany doesn't have the otonal or utonal pentads of the
pentadekanies.